Examples of bounded/continuous sets and functions

Give an example of a set X and a function f: X -> Real numbers such that:

1) f is continuous at 2 points but is discontinuous everywhere else on X.

I have put this:

f(x)=

x^2-x where x belongs to rationals

0 where x doesn't belong to rationals on X= real numbers

Is this right?

2) f is continuous on X and sup {f(x)|x belongs to X} belongs to {f(x)|x belongs to X} but inf{f(x)|x belongs to X} doesn't belong to {f(x)|x belongs to X}

Could this be a possible answer? :

f(x) = x^2 on X = (-10,10]

3) f is bounded on X but not continuous on X

I'm not sure what to put here.

Re: Examples of bounded/continuous sets and functions

Quote:

Originally Posted by

**CourtneyMoon** Give an example of a set X and a function f: X -> Real numbers such that:

1) f is continuous at 2 points but is discontinuous everywhere else on X.

I have put this:

f(x)=x^2-x where x belongs to rationals

0 where x doesn't belong to rationals on X= real numbers

That will work.

Quote:

Originally Posted by

**CourtneyMoon** Give an example of a set X and a function f: X -> Real numbers such that:

2) f is continuous on X and sup {f(x)|x belongs to X} belongs to {f(x)|x belongs to X} but inf{f(x)|x belongs to X} doesn't belong to {f(x)|x belongs to X}

Could this be a possible answer? :

on

The range of your is

Does the range contain both if

Try on

Quote:

Originally Posted by

**CourtneyMoon** Give an example of a set X and a function f: X -> Real numbers such that:

3) f is bounded on X but not continuous on X

Let and .

Re: Examples of bounded/continuous sets and functions

Quote:

Originally Posted by

**Plato** That will work.

The range of your

is

Does the range contain both if

Try

on

Let

and

.

Okay thanks, would f(x)= -x^2 on X =(-1,1) (which you said) work on X=(-1,1] too?

For the f is bounded on X but not continuous on X could I possibly have this:

f(x)= mod x on X = [-10,10] ?

Re: Examples of bounded/continuous sets and functions

Quote:

Originally Posted by

**CourtneyMoon** Okay thanks, would f(x)= -x^2 on X =(-1,1) (which you said) work on X=(-1,1] too?

NO! It does not work for .

For now the range is which contains the .

Quote:

Originally Posted by

**CourtneyMoon** For the f is bounded on X but not continuous on X could I possibly have this: f(x)= mod x on X = [-10,10] ?

What does f(x)= mod x mean?

Re: Examples of bounded/continuous sets and functions

Hmm okay, thank you.

Sorry I meant f(x)=|x| on X = [-10,10]

Re: Examples of bounded/continuous sets and functions

Quote:

Originally Posted by

**CourtneyMoon** Hmm okay, thank you.

Sorry I meant f(x)=|x| on X = [-10,10]

NO again! The function **is continuous everywhere.**

The function I suggested **is continuous nowhere **on .